Urgent. Need them quick

by sealight2107, May 9, 2025, 4:58 PM

With $a,b,c>1$ and $a+b+c=2abc$ . Prove that:
$\sqrt[3]{ab-1}+\sqrt[3]{bc-1}+\sqrt[3]{ca-1} \le \sqrt[3]{(a+b+c)^2}$

inequalities

Divisibilty...

by Sadigly, May 9, 2025, 3:47 PM

Find all $4$ consecutive even numbers, such that the square of their product divides the sum of their squares.

This post has been edited 1 time. Last edited by Sadigly, 42 minutes ago

Interesting inequality

by imnotgoodatmathsorry, May 9, 2025, 2:55 PM

Let $x,y,z > \frac{1}{2}$ and $x+y+z=3$ .Prove that:
$\sqrt{x^3+y^3+3xy-1}+\sqrt{y^3+z^3+3yz-1}+\sqrt{z^3+x^3+3zx-1}+\frac{1}{4}(x+5)(y+5)(z+5) \le 60$

inequalities

k^2/p for k =1 to (p-1)/2

by truongphatt2668, May 9, 2025, 2:05 PM

Let $p$ be a prime such that: $p = 4k+1$ . Simplify:
$\sum_{k=1}^{\frac{p-1}{2}}\begin{Bmatrix}\dfrac{k^2}{p}\end{Bmatrix}$

number theory

Quadratic Residues

Divisibility..

by Sadigly, May 9, 2025, 7:37 AM

Find all $4$ consecutive even numbers, such that the square of their product is divisible by the sum of their squares.

This post has been edited 3 times. Last edited by Sadigly, 44 minutes ago

number theory

AZE JUNIOR NMO

Inspired by Kosovo 2010

by sqing, May 9, 2025, 3:56 AM

Let $a,b>0 , a+b\leq k$ . Prove that
$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq\left(1+\frac{4}{k(k+2)}\right)^2$ $\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq\left(1+\frac{2}{k+2}\right)^2$ Let $a,b>0 , a+b\leq 2$ . Prove that
$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq \frac{9}{4}$ $\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq \frac{9}{4}$

This post has been edited 1 time. Last edited by sqing, Today at 4:04 AM

inequalities proposed

algebra

inequalities

IMO Genre Predictions

by ohiorizzler1434, May 3, 2025, 6:51 AM

Everybody, with IMO upcoming, what are you predictions for the problem genres?

Personally I predict: predict

every lucky set of values {a_1,a_2,..,a_n} satisfies a_1+a_2+...+a_n >n2^{n-1}

by parmenides51, Dec 19, 2020, 1:23 AM

Let $n>1$ be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$ , where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called lucky, if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value).
(a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $a_1+ a_2+...+ a_n < n \cdot 2^n.$ (b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$
Proposed by Ilya Bogdanov

This post has been edited 3 times. Last edited by parmenides51, Dec 20, 2020, 5:11 PM

algebra

Number Theory

by VicKmath7, Mar 17, 2020, 7:31 AM

Let $p$ prime and $m$ a positive integer. Determine all pairs $( p,m)$ satisfying the equation: $p(p+m)+p=(m+1)^3$

number theory

Some really bad rings

by math_explorer, Sep 26, 2017, 12:36 AM

Consider $\mathbb{Q}[x_1, x_2, x_3, \ldots]$ where you adjoin infinitely many free variables. This has infinite Krull dimension because ideals $(x_1, x_2, \ldots, x_n)$ are all prime. It's also local; the ideal generated by all the variables is the unique maximal ideal.

Consider $\mathbb{Q}[x, x^{\omega-n}\text{ for all }n \in \mathbb{Z}]$ ; think of $\omega$ like an infinity. It is local because $(x)$ is the unique maximal ideal. The ideal generated by all $x^{\omega-n}$ is prime; that maximal ideal $(x)$ divides it infinitely many times.

Consider $\mathbb{Z} + x\mathbb{Q}[x]$ , the ring of polynomials with integer constant coefficient. Each integer prime or prime integer or something $p$ generates a maximal ideal $(p)$ . The intersection of those infinitely many maximal ideals is the prime ideal $(x)$ .

algebra

abstract algebra

algebra is hard

by math_explorer, Jul 6, 2017, 4:23 AM

http://mit.edu/bpchen/Public/2017/reconstruction.png

algebra

x+y in B iff x,y in A

by fattypiggy123, Dec 20, 2014, 5:59 AM

Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:

i) $|A| = n$ , $|B| = m$ and $A$ is a subset of $B$
ii) For any distinct $x,y \in B$ , $x+y \in B$ iff $x,y \in A$

Determine the minimum value of $m$ .

absolute value

combinatorics proposed

combinatorics

Submit

how do you have so many posts
by krithikrokcs, Jul 14, 2023, 6:20 PM
lol⠀⠀⠀⠀⠀
by math_explorer, Jan 20, 2021, 8:43 AM
woah ancient blog
by suvamkonar, Jan 20, 2021, 4:14 AM
https://artofproblemsolving.com/community/c47h361466
by math_explorer, Jun 10, 2020, 1:20 AM
when did the first greed control game start?
by piphi, May 30, 2020, 1:08 AM
ok..........
by asdf334, Sep 10, 2019, 3:48 PM
There is one existing way to obtain contributorship documented on this blog. See if you can find it.
by math_explorer, Sep 10, 2019, 2:03 PM
SO MANY VIEWS!!!
PLEASE CONTRIB

by asdf334, Sep 10, 2019, 1:58 PM
Hullo bye
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Hullo bye
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Hullo bye
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It's sad; the blog is still active but not really ;-;
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dope css
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shouts make blogs happier
by briantix, Mar 18, 2016, 9:58 PM
I like your CSS. Can you send me your shoutbox block? I'll credit you.
by bluecarneal, Feb 21, 2011, 8:24 PM
Hey contrib?
by Dsquared, Dec 16, 2010, 3:14 PM
$\sum_{n=0}^{\infty}2^n=-1$
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Delete the shouts...

If you wish something more mathematical, here's some digits.
1679350278936041982752
Enjoy. (no offense)
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Now I feel guilty my shoutbox is so nonmathematical.
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Changed your avatar?
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Gosh, post count over 9000.
by math_explorer, Sep 20, 2010, 1:44 PM
Nice blog I read pretty much whenever you make a new post, but I don't comment.
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I see, that part of the CSS needs a lot of tweaking.
by math_explorer, Sep 12, 2010, 9:20 AM
Hello. Shout?
by asf, Sep 11, 2010, 5:18 PM

91 shouts

math_exploration

by sealight2107, May 9, 2025, 4:58 PM

by Sadigly, May 9, 2025, 3:47 PM

by imnotgoodatmathsorry, May 9, 2025, 2:55 PM

by truongphatt2668, May 9, 2025, 2:05 PM

by Sadigly, May 9, 2025, 7:37 AM

by sqing, May 9, 2025, 3:56 AM

by ohiorizzler1434, May 3, 2025, 6:51 AM

by parmenides51, Dec 19, 2020, 1:23 AM

by VicKmath7, Mar 17, 2020, 7:31 AM

by math_explorer, Sep 26, 2017, 12:36 AM

by math_explorer, Jul 6, 2017, 4:23 AM

by fattypiggy123, Dec 20, 2014, 5:59 AM